The Mpemba effect — hotter cools faster — resists clean explanation partly because the phrase “further from equilibrium” hides a dimensional collapse. Distance from equilibrium is a scalar. The state that is far from equilibrium occupies a specific direction in the space of all possible deviations. Whether the system returns quickly or slowly depends on the direction, not the distance.
Dabelow and Bhattacharyya (2026, arXiv:2603.11707) demonstrate the Mpemba effect entirely within the linear-response regime. No large perturbations, no nonlinear dynamics, no quasiparticle physics. A many-body classical system near equilibrium, evolving under linearized equations of motion. The mechanism: every perturbation decomposes into relaxation eigenmodes, each decaying at its own rate. If the initial state projects strongly onto the fastest-decaying mode and weakly onto the slowest, the system relaxes quickly — regardless of how far from equilibrium it started. A larger perturbation that avoids the slow mode beats a smaller perturbation that excites it.
The spectral separation between fast and slow modes is the load-bearing structure. When the gap is large, initial conditions that happen to miss the slow mode have a dramatic speed advantage. The effect is not exotic. It is a direct consequence of multi-exponential relaxation with well-separated timescales — a feature of nearly every dissipative physical system.
Tang, Katsura, and Bhatt (2026, arXiv:2603.11788) find the same structure in quantum many-body systems through a different route: Ruelle-Pollicott resonances. The quantum Mpemba effect occurs when a state farther from equilibrium has suppressed overlap with the dominant resonant mode. Translation-symmetry breaking provides the mechanism — it reshapes which resonances the initial state excites without changing the distance from equilibrium.
The quantum case makes the geometry literal. The Hilbert space of the system has a basis of resonant modes, each with a decay rate. The initial state is a vector in this space. Whether relaxation is anomalously fast depends on the angle between this vector and the slow-decaying subspace — a geometric property of the initial condition, not a thermodynamic one.
This reframes the existing understanding. The quantum Mpemba effect was first observed in quench experiments where a larger symmetry-breaking tilt produced faster symmetry restoration. The explanation (Joshi, Franke et al., PRL 133, 2024) was that larger deviations generate faster quasiparticles — the deviation literally creates the engine of its own correction. That mechanism is real but specific to quench dynamics far from equilibrium.
The linear-regime results show the effect doesn't need the engine. Near equilibrium, there are no fast quasiparticles. There is only the spectral decomposition of the perturbation and the separation of decay rates. The “engine” interpretation is a special case of the general structure: the deviation happened to point in a direction that projected onto fast modes. Far from equilibrium, the magnitude of the deviation created the favorable projection. Near equilibrium, the projection is there or it isn't, independent of magnitude.
The through-claim: distance from equilibrium is the wrong coordinate for predicting relaxation speed. The right coordinate is the angle in mode space — the projection of the initial perturbation onto slow versus fast eigenmodes. This is a geometric statement, not a thermodynamic one, and it explains why the Mpemba effect appears across classical and quantum systems, near and far from equilibrium, in systems with wildly different microscopic dynamics. The geometry of relaxation is universal. The mechanisms that produce favorable projections are particular. The same lesson applies anywhere multi-timescale relaxation occurs: the system that starts further from the target arrives first whenever its displacement points away from the slow direction. Whether the displacement was caused by a thermal quench, a symmetry-breaking tilt, or a gentle push near equilibrium is irrelevant. What matters is not how far you are from where you're going, but which modes you've excited along the way.